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Most Class 7 students can learn the rules for multiplying and dividing fractions in an afternoon. The harder part is knowing what the answer should look like before you calculate it. This post walks through Chapter 8 of the new maths book, with worked examples, the mistakes that cost marks, and a short practice routine.
Which chapter is this?
NCERT's Class 7 maths book is Ganita Prakash, Grade 7 Part 1. Its introduction says Chapter 8, "Working with Fractions", builds on what students know about fractions and explores multiplying and dividing them. The Chapter 8 PDF has three sections: multiplication of fractions, division of fractions, and problems involving fractions. Fractions themselves, including adding and subtracting them, are introduced in Chapter 7 of the Class 6 book.
How do you multiply two fractions?
The book uses a unit square as the whole. A rectangle with sides 1/2 and 1/4 inside it fits 8 times into the square, so its area, and the product 1/2 × 1/4, is 1/8. The area of a rectangle with fractional sides equals the product of its sides, and that picture leads to the rule
a/b × c/d = (a × c)/(b × d).
The chapter notes that Brahmagupta stated this rule in this general form in his *Brāhmasphuṭasiddhānta* in 628 CE. A whole number works too: write it with denominator 1, so 3 × 3/4 = 3/1 × 3/4 = 9/4.
Worked example 1. A tap fills 7/10 of a tank in 1 hour. How much does it fill in 2/3 of an hour? (This is from the chapter's exercises.)
7/10 × 2/3 = 14/30. Divide top and bottom by 2: 7/15 of the tank.
Worked example 2: cancel first. 14/15 × 25/42.
14 and 42 share a factor of 14: 14 becomes 1, 42 becomes 3.
25 and 15 share a factor of 5: 25 becomes 5, 15 becomes 3.
Now multiply: (1 × 5)/(3 × 3) = 5/9.
Check without cancelling: 14 × 25 = 350 and 15 × 42 = 630, and 350/630 divided by 70 on top and bottom is 5/9. Same answer, smaller numbers.
Why can multiplying make a number smaller?
The chapter asks when a product is bigger than both numbers. With whole numbers it always is (3 × 5 = 15). With fractions less than 1 it is not: 1/2 × 1/3 = 1/6, which is smaller than both. Multiplying by a fraction less than 1 means taking a part of something, so you get less.
This is where many students go wrong. In a study in the Journal of Educational Psychology (Siegler and Lortie-Forgues, 2015), middle school students and trainee teachers were asked to predict, without calculating, whether an answer would be larger or smaller. On multiplication and division with fractions below 1, they were right less often than chance, even though many of them could do the calculation correctly.
How do you divide by a fraction?
The book turns division into multiplication. 12 ÷ 4 = 3 because 4 × 3 = 12. In the same way, 6 ÷ 1/4 asks "how many quarters make 6?" There are 4 quarters in each whole, so 6 × 4 = 24 quarters. The quotient, 24, is bigger than 6.
The number you multiply by to get 1 is the reciprocal. The reciprocal of 3/5 is 5/3, because 3/5 × 5/3 = 1. So the rule, which the book also credits to Brahmagupta, is:
a/b ÷ c/d = a/b × d/c.
Worked example 3. 2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9. Is that sensible? You are dividing by a number below 1, so the answer should be bigger than 2/3. It is: 10/9 is more than 1.
Worked example 4: a problem from 860 CE. The chapter quotes a puzzle from Pṛithūdakasvāmī's commentary on Brahmagupta. Four fountains fill a cistern in 1 day, 1/2 day, 1/4 day and 1/5 day. How long do they take together?
In one day, the first fills it 1 ÷ 1 = 1 time.
The second fills it 1 ÷ 1/2 = 2 times.
The third fills it 1 ÷ 1/4 = 4 times.
The fourth fills it 1 ÷ 1/5 = 5 times.
Together: 1 + 2 + 4 + 5 = 12 times a day, so one filling takes 1/12 of a day, which is 24 ÷ 12 = 2 hours.
What mistakes do Class 7 students make with fractions?
Flipping the wrong fraction. Take the reciprocal of the divisor (the second number), never the dividend.
Cancelling before flipping in a division. In 4/9 ÷ 2/3, flip first: 4/9 × 3/2. Only then cancel, giving 2/3.
Using cancelling in addition. Cancelling across fractions only works for multiplication. 1/2 + 2/4 is not 1/4 + 1/4 by "cancelling".
Multiplying mixed numbers part by part. 2½ × 1½ is not (2 × 1) + (½ × ½) = 2¼. Convert first: 5/2 × 3/2 = 15/4 = 3¾.
Not checking size. If your answer to a "part of" question is bigger than the whole, something has gone wrong.
A practice routine for the week
Day 1: draw unit-square pictures for 1/2 × 1/3, 2/3 × 3/4 and 3/5 × 1/2, then check them with the rule.
Day 2: do ten multiplications, cancelling first, and check two of them the long way.
Day 3: before ten divisions, write "bigger" or "smaller" next to each, then solve and see how many predictions were right.
Day 4: solve two word problems from the chapter's Section 8.3, and write one of your own about cooking or sharing.
Day 5: mix all types in one set without headings, the way an exam will.
If you get stuck on why a step works, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than just printing an answer. Parents who want to help without taking over can use the ideas in our post on helping with word problems.
Start today: write "bigger" or "smaller" beside five fraction sums before you solve any of them.
